#### Awardees for Master Theses

Note: All prizes and awards are subjected to the awardee's presence at the Award Ceremony in 2010. The tentative date for the ceremony is December 17, 2010.

Gold Awards

Name: Sz-Sheng Wang(ÍõÙnÂ})

Title: Extensions of multiply twisted pluri-canonical forms

Affiliation: National Taiwan University

Abstract: Using Siu¡¯s version of the Ohsawa-Takegoshi theorem and the techniques in proving the invariance of plurigenera developed by Siu and Paun, we show in Theorem 1.1 that for a smooth projective family pluricanonical forms with integrable pseudo-effective twisted coefficients may be lifted from the central fiber to the ambient family.

Our result can be regarded as certain ¡°tensor product version¡± of the Ohsawa-Takegoshi extension theorem. It unifies and generalizes most of the known extension results for smooth projective families.

Silver Awards (No order)

It is decided that the following three persons will be awarded the title of Silver Awardees in the 2010 ceremony. They will equally share the sum of the two announced silver award prizes.

Name: Yun Kuen Cheung

Title: Analysis of Weighted Digital Sums by Mellin Transform

Affiliation: Hong Kong University of Science and Technology

Abstract: In this thesis, we analyze two types of weighted digital sums (WDS) which arise in algorithm analysis.¡­¡­

Title: Spectral asymptotics for large skew-symmetric perturbations of the harmonic oscillator and generalizations

Affiliation: Universit¨¦ Pierre et Marie Curie

Abstract:Motivated by a stability problem in fluid mechanics, we are interested in the spectral and pseudospectral properties of the differential operator $H_\epsilon=-\partial_x^2+x^2+i\epsilon^{-1}f(x)$ on $L^2(\R)$, where $f$ is a real-valued function and $\epsilon>0$ is a small parameter. Under appropriate conditions on $f$, I.Gallagher, T.Gallay and F.Nier gave precise estimates by using localization techniques and semiclassical subelliptic estimates. This master thesis is divided into two parts. Section 1 is a description of the work in the article I.Gallagher, T.Gallay and F.Nier, Section 2 exhibits generalizations to dimension $n$ of the resolvent estimates of the operator $H_\epsilon$ along the imaginary axis.

Name: Fan Sin Tsun

Title: Open Orbits and Augmentations of Dynkin Diagrams

Affiliation: Chinese University of Hong Kong

Abstract: The study of open orbits in finite dimensional representations has its roots in geometry, representation theory and invariant theory. Our motivation starts from considering the open GLn-orbits in ¦«kRn as a local model for defining the stable forms on smooth manifolds. The observation is that such an open GLn ¨Corbit exists in ¦«kRn exactly when the Dynkin diagram of slnC, i.e. of type An-1, can be extended to another Dynkin diagram by attaching an extra node to the k-th node of the original diagram. In this thesis, we will fully elaborate this idea using techniques in representation theory. We show that an augmentation of Dynkin diagrams provides a structure of Z-gradation on the ambient Lie algebra corresponding to the larger Dynkin diagram and the 0-th graded piece is then a reductive Lie subalgebra whose semisimple part is associated to the smaller Dynkin diagram. Passing to the group level, we obtain an irreducible representation which admits a finite number of orbits, which will be called an irreducible prehomogeneous vector space of parabolic type. We also consider other cases of irreducible representations which admit an open orbit but not coming from an augmentation of Dynkin diagrams. Finally, we mention a generalization of prehomoegeneous vector spaces of parabolic type to the visible representations motivated from invariant theory and a treatment of the real forms of irreducible prehomogeneous vector spaces of parabolic type.